How to understand the off-diagonal elements

Main Question or Discussion Point

In quantum mechanics, observable variables are represented by operators, and thus can be replaced by matrix in a certain basis.
If we have H|n>=E(n)|n>, where |n> are eigenfunctions of Hamilton matrix. Here is the problem: what's the physical meaning of <m|P|n>, namely off-diagonal elements of another observable operator, such as momentum?
I have known that sometimes this has something to do with quantum transition. But I want to know is there any universal understanding of off-diagonal elements?
Thanks!

what's the physical meaning of <m|P|n>, namely off-diagonal elements of another observable operator, such as momentum?
I have known that sometimes this has something to do with quantum transition. But I want to know is there any universal understanding of off-diagonal elements?
Thanks!

Many operators can be interpreted as generators of infinitesimal transformations on states. For example consider the momentum operator ##P##. If I have a state ##| \psi \rangle## and ##\epsilon## is an infinitesimal distance, then ##(1 + i \frac{\epsilon}{\hbar} P ) | \psi \rangle## is a version of the state ##| \psi \rangle## that has been translated to the left by a distance ##\epsilon##.

Then ##\langle m | P | n \rangle## can be interpreted as something like "the change in the overlap of ##| m \rangle## and ##| n \rangle## when one of them is translated by a small distance."